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Question Number 202109 by hardmath last updated on 20/Dec/23

$$$$How many different three-digit numbers a satisfy the condition GCD(a;18)>=2 ?

Answered by mr W last updated on 22/Dec/23

$$100⩽a⩽999$$$$18=2\times 3^2$$$$18hasfollowingfactorswhichare$$$$equalorgreaterthan2:$$$$2,3,6,9,18$$$$thatmeansamustbeamultipleof$$$$2or3or6or9or18.$$$$a=2k:\Rightarrow \lfloor \frac{999}{2}\rfloor -\lfloor \frac{99}{2}\rfloor =499-49=450$$$$a=3k:\Rightarrow \lfloor \frac{999}{3}\rfloor -\lfloor \frac{99}{3}\rfloor =333-33=300$$$$a=6k:\Rightarrow \lfloor \frac{999}{6}\rfloor -\lfloor \frac{99}{6}\rfloor =166-16=150$$$$a=9k:\Rightarrow \lfloor \frac{999}{9}\rfloor -\lfloor \frac{99}{9}\rfloor =111-11=100$$$$a=18k:\Rightarrow \lfloor \frac{999}{18}\rfloor -\lfloor \frac{99}{18}\rfloor =55-5=50$$$$$$$$18k9k6k3k2k$$$$50+100+150+300+450$$$$50+\cancel{\underset{50}{100}}+\cancel{\underset{100}{150}}+\cancel{\underset{\cancel{\underset{\cancel{\underset{100}{200}}}{250}}}{300}}+\cancel{\underset{\cancel{\underset{}{400}}}{450}}$$$$50+50+100+100+300=600numbers✓$$

Commented by hardmath last updated on 21/Dec/23

$$\mathrm{perfect}\mathrm{dear}\mathrm{professor},\mathrm{thank}\mathrm{you}\mathrm{so}\mathrm{much}$$

Commented by hardmath last updated on 21/Dec/23

$$$$My dear professor, excuse me, what does the red part on the last line mean? And where did they come from?

Commented by mr W last updated on 22/Dec/23

$$ifanumberisdivisibleby18,then$$$$itisalsodivisibleby9,by2,by6,by3.$$$$similarlyifanumberisdivisible$$$$by9,thenitisalsodivisibleby3.$$$$andifitisdivisibleby6,itisalso$$$$divisibleby3andby2.$$$$sowemustsubstractthenumbers$$$$whicharedoublycountedasshown$$$$inthelastline.$$

Commented by hardmath last updated on 22/Dec/23

$$\mathrm{Excellent}\mathrm{dear}\mathrm{pfofessor},$$$$\mathrm{thank}\mathrm{you}\mathrm{very}\mathrm{mych}$$

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